Finite Group Databasebeta

← #7623134

Subgroups — #7623134

Order 512.

1550 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60718C211yesmaximal, characteristic, complemented
2C4×#576C211yesmaximal, characteristic, complemented
3#33684C214yesmaximal, complemented
4#33678C214yesmaximal, complemented
5#33675C212yesmaximal, complemented
6#33670C212yesmaximal, complemented
7#33667C212yesmaximal, complemented
8#33660C212yesmaximal, complemented
9#33652C212yesmaximal, complemented
10#33629C211yesmaximal, characteristic, complemented
11C2×#2224C211yesmaximal, characteristic, complemented
12#33580C211yesmaximal, characteristic, complemented
13#33579C211yesmaximal, characteristic, complemented
14#33526C212yesmaximal, complemented
15#33459C212yesmaximal, complemented
16#33454C212yesmaximal, complemented
17#33423C211yesmaximal, characteristic, complemented
18C2×#576C2211yescharacteristic
19#3511—21no—
20C4×H(5,𝔽2)—21no—
21C2×#509C2212yes—
22C2×#508C2212yes—
23#3182—22no—
24#3175—22no—
25#3175C2214yescomplemented
26#3170C2212yescomplemented
27#3169C2212yescomplemented
28C2×#500C2211yescharacteristic, complemented
29#3147C2211yescharacteristic, complemented
30#3141—21no—
31#3061—21no—
32#3061C2212yes—
33#2951—21no—
34#2951C2212yes—
35#2951C2211yescharacteristic
36#2951C2212yescomplemented
37#2948C2212yes—
38#2948C2211yescharacteristic
39#2946—22no—
40#2946—21no—
41C4×AGL(1,ℤ/8)—21no—
42C4×AGL(1,ℤ/8)—22no—
43C4×#135C2211yescharacteristic
44C4×#135C2212yes—
45C2×C4×D8C2211yescharacteristic
46C2×C4×D8C2212yes—
47#2913C2211yescharacteristic
48#2897—21no—
49C4×#131C2211yescharacteristic
50#2237C2214yescomplemented
51#2236C2214yescomplemented
52#2234C2214yescomplemented
53#2232C2214yescomplemented
54#2227C2212yescomplemented
55#2226C2212yescomplemented
56#2225C2212yescomplemented
57#2224C2212yescomplemented
58#2224C2212yescomplemented
59#2224C2214yescomplemented
60#2224C2212yescomplemented
61#2211C2212yescomplemented
62#2210C2212yescomplemented
63#2210C2214yescomplemented
64#2209C2211yescharacteristic, complemented
65#2208C2211yescharacteristic, complemented
66#2208C2212yescomplemented
67#2207C2211yescharacteristic, complemented
68#2206C2211yescharacteristic, complemented
69#2206C2212yescomplemented
70#2189C2214yes—
71#2187C2214yes—
72#2186C2214yes—
73#2184C2214yes—
74#2177C2214yescomplemented
75#2175C2214yescomplemented
76#2175C2214yescomplemented
77#2175C2214yescomplemented
78C4×Q32C2214yescomplemented
79C4×SD32C2214yescomplemented
80C4×D16C2214yescomplemented
81C4×D16C2214yescomplemented
82#2164C2214yescomplemented
83#2163C2212yescomplemented
84#2162C2212yescomplemented
85#2159C2212yescomplemented
86C2×#366C2212yescomplemented
87#2150C2211yescharacteristic, complemented
88C2×C4⋊C16C2211yescharacteristic, complemented
89#2139C2212yes—
90#2139C2212yescomplemented
91#2138C2212yescomplemented
92C2×#357C2212yes—
93#2113C2212yes—
94#2108C2212yescomplemented
95C2×H(5,𝔽2)—21no—
96#576—24no—
97#576—24no—
98C2×AGL(1,ℤ/8)—21no—
99C2×AGL(1,ℤ/8)—22no—
100C2×#135—21no—
101C2×#135C2312yes—
102C2×#135C2312yes—
103C2×#135C2311yescharacteristic
104C22×D8C2312yes—
105C22×D8C2311yescharacteristic
106C2×#131D411yescharacteristic
107C2×#131C2311yescharacteristic
108#556—41no—
109#555—41no—
110#554—41no—
111#553—42no—
112#548—42no—
113#547—41no—
114#542—24no—
115#537—22no—
116#533—22no—
117#532—22no—
118#532—21no—
119C2×C4⋊Q8—21no—
120#520—21no—
121#520—21no—
122#520—22no—
123#520—22no—
124#520—41no—
125#518—21no—
126#518—22no—
127#518—42no—
128#518—41no—
129C4×#41—21no—
130C4×#41—22no—
131C4×#41—41no—
132C4×#41D411yescharacteristic
133C2×C4×D4—21no—
134C2×C4×D4—22no—
135C2×C4×D4—41no—
136C2×C4×D4—42no—
137#509—28no—
138#508—28no—
139C2×Q32C2312yes—
140C2×SD32C2312yes—
141C2×SD32C2312yes—
142C2×SD32C2314yes—
143C2×D16C2312yes—
144C2×D16C2314yes—
145C2×M32C2312yes—
146C22×C16C2312yes—
147#500—21no—
148#500—21no—
149#500C2312yes—
150#500C2312yes—
151#495—21no—
152#495C2312yes—
153#494C2311yescharacteristic
154#493C2311yescharacteristic
155#492—21no—
156#489—24no—
157#487—22no—
158#486—22no—
159#468—24no—
160#468—24no—
161#467—22no—
162#467C2314yes—
163#466—22no—
164#466C2314yes—
165#442—22no—
166#442—22no—
167#442—22no—
168#440—24no—
169#440—24no—
170#440—22no—
171#440—22no—
172#440—22no—
173C4×Q16—22no—
174C4×Q16C2314yes—
175C4×Q16C2311yescharacteristic
176C4×Q16C2312yes—
177C4×Q16C2311yescharacteristic
178C4×SD16—22no—
179C4×SD16—21no—
180C4×SD16—22no—
181C4×SD16—21no—
182C4×D8—22no—
183C4×D8—22no—
184C4×D8—21no—
185C4×D8—22no—
186C4×D8—21no—
187C4×D8C2314yes—
188C4×D8C2312yes—
189C4×D8C2311yescharacteristic
190C4×D8C2312yes—
191C4×D8C2312yes—
192C4×D8C2312yes—
193C4×D8C2312yes—
194C4×D8C2311yescharacteristic
195#428—21no—
196#428—22no—
197#427C2311yescharacteristic
198#427C2312yes—
199C2×#107—21no—
200C2×#107C2312yes—
201C2×#107C2311yescharacteristic
202C2×#107C2312yes—
203#418—21no—
204#418—22no—
205#417—22no—
206#417—22no—
207#417—21no—
208#417—22no—
209#416—21no—
210#416—22no—
211C2×#102—21no—
212C2×#102—22no—
213#405D412yes—
214#405D411yescharacteristic
215#405C2311yescharacteristic
216#405C2312yes—
217C4×M16D411yescharacteristic
218C4×M16D412yes—
219C2×C4×C8C2311yescharacteristic
220C2×C4×C8C2312yes—
221#367C2314yes—
222#366C2314yes—
223#366C2314yes—
224#366C2314yes—
225C4⋊C16C2311yescharacteristic
226C4⋊C16C2311yescharacteristic
227C4⋊C16C2312yes—
228C4⋊C16C2312yes—
229C4⋊C16C2312yes—
230#358C2312yes—
231#358C2312yes—
232#358C2314yes—
233#358C2314yes—
234#357C2314yes—
235#357C2314yes—
236#357C2312yes—
237#357C2312yes—
238#357C2314yes—
239#357C2314yes—
240C22⋊C16C2314yes—
241C22⋊C16C2314yes—
242#346C2314yes—
243C4×C16C2314yes—
244H(5,𝔽2)—28no—
245H(5,𝔽2)—44no—
246C2×#41—22no—
247C2×#41—21no—
248C2×#41—22no—
249C2×#41—41no—
250C2×#41—42no—
251C2×#41C2×D411yescharacteristic
252C22×Q8—41no—
253C22×D4—22no—
254C22×D4—21no—
255C22×D4—42no—
256C22×D4—41no—
257C23×C4—41no—
258C23×C4—42no—
259AGL(1,ℤ/8)—28no—
260AGL(1,ℤ/8)—216no—
261#135—24no—
262#135—24no—
263#135—22no—
264#135—24no—
265#135—24no—
266#135—44no—
267#135#4114yescomplemented
268C2×Q16—21no—
269C2×Q16C2412yes—
270C2×Q16C2411yescharacteristic
271C2×SD16—22no—
272C2×SD16—21no—
273C2×SD16—22no—
274C2×SD16—22no—
275C2×D8—24no—
276C2×D8—24no—
277C2×D8—22no—
278C2×D8—21no—
279C2×D8—21no—
280C2×D8—22no—
281C2×D8#4114yescomplemented
282C2×D8C2412yes—
283C2×D8C2411yescharacteristic
284C2×D8C2412yes—
285#131—22no—
286#131—22no—
287#131—42no—
288C2×M16C2×D411yescharacteristic
289C2×M16C2×D411yescharacteristic
290C2×M16C2×D412yes—
291C2×M16C2×D412yes—
292C22×C8C2×D411yescharacteristic
293C22×C8C2×D412yes—
294C22×C8C2412yes—
295C22×C8C2411yescharacteristic
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—21no—
299C4⋊Q8—41no—
300C4⋊Q8—41no—
301#127—21no—
302#126—44no—
303#125—42no—
304#124—22no—
305#124—44no—
306#124—42no—
307#123—44no—
308C22⋊Q8—24no—
309C22⋊Q8—42no—
310C22⋊Q8—42no—
311C22⋊Q8—44no—
312#121—24no—
313#121—42no—
314#121—44no—
315#121—41no—
316#121—41no—
317#121—42no—
318C4×Q8—21no—
319C4×Q8—22no—
320C4×Q8—21no—
321C4×Q8—24no—
322C4×Q8—41no—
323C4×Q8—42no—
324C4×Q8D811yescharacteristic
325C4×Q8D811yescharacteristic
326C4×D4—22no—
327C4×D4—21no—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4—22no—
332C4×D4—21no—
333C4×D4—24no—
334C4×D4—42no—
335C4×D4—41no—
336C4×D4—41no—
337C4×D4—42no—
338C4×D4—41no—
339C4×D4—42no—
340C4×D4—41no—
341C4×D4—42no—
342C4×D4—41no—
343C4×D4—42no—
344C4×D4—42no—
345C4×D4—42no—
346C4×D4—42no—
347C4×D4—44no—
348C4×D4—41no—
349C4×D4—42no—
350C4×D4—42no—
351C4×D4—41no—
352C4×D4—41no—
353C4×D4—42no—
354C4×D4—41no—
355C4×D4D811yescharacteristic
356C4×D4D812yes—
357C4×D4D812yes—
358C4×D4D811yescharacteristic
359#117—22no—
360#117—21no—
361#117—42no—
362#117—41no—
363#117—42no—
364#117—42no—
365#117—42no—
366#117—41no—
367#117C2×D411yescharacteristic
368#117C2×D412yes—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—41no—
373C2×C4⋊C4—41no—
374C2×C4⋊C4—42no—
375C2×C4⋊C4—42no—
376C2×C22⋊C4—41no—
377C2×C22⋊C4—42no—
378C2×C22⋊C4—42no—
379C2×C22⋊C4—42no—
380C2×C22⋊C4—42no—
381C2×C42—41no—
382C2×C42—42no—
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D412yes—
385Q32—24no—
386SD32—24no—
387SD32—24no—
388SD32—28no—
389D16—24no—
390D16—28no—
391M32—24no—
392C2×C16—24no—
393C2×C16C2412yes—
394C2×C16C2414yes—
395C2×C16C2412yes—
396#107—22no—
397#107—22no—
398#107—22no—
399#107—21no—
400#107—22no—
401#107C2414yes—
402#107C2411yescharacteristic
403#107C2412yes—
404#107C2411yescharacteristic
405#106—22no—
406#106—21no—
407#103—21no—
408#103—22no—
409#103—21no—
410#103—24no—
411#103—24no—
412#102—22no—
413#102—21no—
414#102—22no—
415#102—22no—
416#102—22no—
417#102—22no—
418#102—21no—
419#102—24no—
420#102—24no—
421#97C2×D412yes—
422#97C2×D412yes—
423#97C2×D412yes—
424C4×C8C2×D412yes—
425C4×C8C2×D411yescharacteristic
426C4×C8C2×D412yes—
427C4×C8C2×D411yescharacteristic
428C4×C8C2412yes—
429C4×C8C2411yescharacteristic
430C4×C8C2411yescharacteristic
431C24—42no—
432C24—41no—
433#41—22no—
434#41—22no—
435#41—28no—
436#41—24no—
437#41—44no—
438#41—48no—
439#41—42no—
440#41—44no—
441#41—44no—
442#41—164no—
443C2×Q8—21no—
444C2×Q8—22no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—162no—
448C2×Q8C2×D811yescharacteristic
449C2×D4—28no—
450C2×D4—24no—
451C2×D4—21no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—44no—
455C2×D4—416no—
456C2×D4—48no—
457C2×D4—42no—
458C2×D4—41no—
459C2×D4—42no—
460C2×D4—44no—
461C2×D4—42no—
462C2×D4—41no—
463C2×D4—41no—
464C2×D4—41no—
465C2×D4—42no—
466C2×D4C2×D812yes—
467C2×D4C2×D811yescharacteristic
468C22×C4—42no—
469C22×C4—41no—
470C22×C4—42no—
471C22×C4—41no—
472C22×C4—42no—
473C22×C4—41no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—41no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—44no—
482C22×C4—162no—
483C22×C4—164no—
484C22×C4C2×D811yescharacteristic
485C22×C4C2×D812yes—
486C22×C4C2×D811yescharacteristic
487C22×C4C2×D812yes—
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490Q16—21no—
491Q16—22no—
492Q16—22no—
493Q16—42no—
494Q16C2×#4112yes—
495SD16—28no—
496SD16—24no—
497SD16—44no—
498SD16—44no—
499D8—28no—
500D8—24no—
501D8—22no—
502D8—21no—
503D8—22no—
504D8—22no—
505D8—42no—
506D8C2×#4112yes—
507D8C2×#4114yes—
508M16—22no—
509M16—22no—
510M16—24no—
511M16—42no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—42no—
516C2×C8C22×D411yescharacteristic
517C2×C8C22×D412yes—
518C2×C8C22×D412yes—
519C2×C8C22×D411yescharacteristic
520C2×C8C22×D411yescharacteristic
521C2×C8C2×#4112yes—
522C2×C8C2×#4112yes—
523C2×C8C2511yescharacteristic
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—24no—
528C4⋊C4—42no—
529C4⋊C4—41no—
530C4⋊C4—42no—
531C4⋊C4—41no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—41no—
537C4⋊C4—41no—
538C4⋊C4—41no—
539C4⋊C4—44no—
540C4⋊C4C2×D811yescharacteristic
541C4⋊C4C2×D812yes—
542C4⋊C4C2×D811yescharacteristic
543C4⋊C4C2×D812yes—
544C22⋊C4—42no—
545C22⋊C4—41no—
546C22⋊C4—44no—
547C22⋊C4—41no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—41no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—42no—
557C22⋊C4—42no—
558C22⋊C4C2×D812yes—
559C22⋊C4C2×D812yes—
560C22⋊C4C2×D812yes—
561C42—41no—
562C42—42no—
563C42—41no—
564C42—42no—
565C42—41no—
566C42—44no—
567C42C2×D811yescharacteristic
568C42C2×D812yes—
569C42C2×D812yes—
570C42C2×D811yescharacteristic
571C42C22×D411yescharacteristic
572C42C22×D412yes—
573C42C22×D411yescharacteristic
574C16—22no—
575C16—22no—
576C16—24no—
577C23—48no—
578C23—42no—
579C23—41no—
580C23—44no—
581C23—42no—
582C23—161no—
583C23—162no—
584C23—161no—
585C23—162no—
586C23C22×D812yes—
587C23C22×D811yescharacteristic
588Q8—22no—
589Q8—22no—
590Q8—41no—
591Q8—42no—
592Q8—42no—
593Q8—84no—
594Q8—162no—
595D4—24no—
596D4—22no—
597D4—22no—
598D4—24no—
599D4—48no—
600D4—48no—
601D4—44no—
602D4—44no—
603D4—42no—
604D4—41no—
605D4—42no—
606D4—42no—
607D4—82no—
608D4—84no—
609D4—82no—
610D4—164no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—48no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—84no—
626C2×C4—84no—
627C2×C4—82no—
628C2×C4—162no—
629C2×C4—161no—
630C2×C4—162no—
631C2×C4—162no—
632C2×C4—162no—
633C2×C4—164no—
634C2×C4—161no—
635C2×C4—162no—
636C2×C4—162no—
637C2×C4#52512yes—
638C2×C4C22×D812yes—
639C2×C4C22×D812yes—
640C2×C4C22×D811yescharacteristic
641C2×C4C22×D811yescharacteristic
642C2×C4C22×D811yescharacteristic
643C2×C4C22×D812yes—
644C2×C4C22×D811yescharacteristic
645C2×C4C22×D811yescharacteristic
646C2×C4#57512yes—
647C2×C4C23×D411yescharacteristic
648C8—22no—
649C8—21no—
650C8—21no—
651C8—22no—
652C8#52512yes—
653C8C22×#4111yescharacteristic
654C8C22×#4111yescharacteristic
655C22—21no—
656C22—21no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—44no—
661C22—42no—
662C22—82no—
663C22—81no—
664C22—82no—
665C22—81no—
666C22—82no—
667C22—82no—
668C22—82no—
669C22—164no—
670C22—168no—
671C22#310012yes—
672C22C23×D811yescharacteristic
673C4—22no—
674C4—22no—
675C4—21no—
676C4—21no—
677C4—22no—
678C4—21no—
679C4—42no—
680C4—82no—
681C4—81no—
682C4—82no—
683C4—84no—
684C4—81no—
685C4#310012yes—
686C4#341512yes—
687C4C2×#52511yescharacteristic
688C4C2×#57511yescharacteristic
689C2—22no—
690C2—42no—
691C2—84no—
692C2—82no—
693C2—162no—
694C2C2×#310011yescharacteristic
695C2C2×#341511yescharacteristic
696C2#6263711yescharacteristic
subgroupG/H
center, socleC22C23×D8
commutator, Frattini, cosocleC2×C8C25
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G
LECSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #60718C2semidirect
2. C4×#576C2semidirect
3. #33684C2semidirect
4. #33678C2semidirect
5. #33675C2semidirect
6. #33670C2semidirect
7. #33667C2semidirect
8. #33660C2semidirect
9. #33652C2semidirect
10. #33629C2semidirect
11. C2×#2224C2semidirect
12. #33580C2semidirect
13. #33579C2semidirect
14. #33526C2semidirect
15. #33459C2semidirect
16. #33454C2semidirect
17. #33423C2semidirect
25. #3175C22semidirect
26. #3170C22semidirect
27. #3169C22semidirect
28. C2×#500C22semidirect
29. #3147C22semidirect
36. #2951C22semidirect
50. #2237C22semidirect
51. #2236C22semidirect
52. #2234C22semidirect
53. #2232C22semidirect
54. #2227C22semidirect
55. #2226C22semidirect
56. #2225C22semidirect
57. #2224C22semidirect
58. #2224C22semidirect
59. #2224C22semidirect
60. #2224C22semidirect
61. #2211C22semidirect
62. #2210C22semidirect
63. #2210C22semidirect
64. #2209C22semidirect
65. #2208C22semidirect
66. #2208C22semidirect
67. #2207C22semidirect
68. #2206C22semidirect
69. #2206C22semidirect
74. #2177C22semidirect
75. #2175C22semidirect
76. #2175C22semidirect
77. #2175C22semidirect
78. C4×Q32C22semidirect
79. C4×SD32C22semidirect
80. C4×D16C22semidirect
81. C4×D16C22semidirect
82. #2164C22semidirect
83. #2163C22semidirect
84. #2162C22semidirect
85. #2159C22semidirect
86. C2×#366C22semidirect
87. #2150C22semidirect
88. C2×C4⋊C16C22semidirect
90. #2139C22semidirect
91. #2138C22semidirect
94. #2108C22semidirect
267. #135#41semidirect
281. C2×D8#41semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#607182
C4×#5762
#336842
#336782
#336752
#336702
#336672
#336602
#336522
#336292
C2×#22242
#335802
#335792
#335262
#334592
#334542
#334232